Of Particular Significance

Chapter 14, Note 6: A Subtlety with Measuring Light’s Speed

It’s easy to measure the speed of something as it moves right by you; you can use a short straight ruler and your own clock to do it. You measure its nearby location at one time, and you measure the location again a second time a short time later, and from those measurements you can infer its speed. Specifically, you determine the distance D between location 2 and location 1, and the time T between measurement 1 and measurement 2, and then the average speed is simply

  • vaverage = D/T

But implicitly, in making this measurement, you have assumed that the space and the time around you are flat and unchanging. That’s why you didn’t have to think very hard when you measured what the locations and the times actually were.

Your assumption is fine if you are measuring things nearby. Around the Earth, it happens that it is a very good assumption out to very great distances. But it often isn’t true in Einstein’s universe.

As Einstein taught us, gravity royally messes with both distances and times. Space can be curved, and its shape can change over time. That can easily make a speed measurement ambiguous. If you have any problem measuring the two locations or of making sense of the distance between those two locations, or if you have trouble determining the times at which you made those measurements or the duration of the time between them, then your speed measurement is doomed.

In particular, it is not true that the speed of faraway light waves, relative to you, is always c. You can’t even define that speed unambiguously. Sometimes, in situations that are neither too simple nor too complex, you can at least try to define the distance between you and a light wave (or something else). If and when you do so, you’ll discover that this distance can change at any rate, slower than c or even faster than c.

A Rapidly Expanding Rubber Sheet

There’s a famous analogue (often confusingly stated) which can give some insight into how this works, although the analogy is not perfect.

If we take a rubber sheet with ants on it, the ants can’t move very quickly and so any two ants that pass each other do so at a low speed.

But if we now pull on the edges of the sheet, so that it expands very rapidly, the ants will move rapidly apart, dragged along with the sheet, much faster than they themselves could walk. Close ants barely notice; any two ants that pass each other will still do so at a walking speed. But distant ants fly apart.

We can see this in the figure below, which shows a stationary square rubber sheet with ants walking on it at a fixed speed. When the sheet begins rapidly to expand, the ants continue to walk, but the expansion drags them along. Nearby ants still move slowly relative to one another (see the lower right of the animation), initially oblivious to the expansion. But the distance between faraway ants increases much more rapidly than they can walk, and if the expansion continues to accelerate, they can never meet again.

Ants move around at walking speed on a rubber sheet. Suddenly a machine stretches the rubber at an ever increasing rate. The distance between distant pairs of ants then grows much faster than they can walk; this is less immediately true for nearby ants. Photons in an inflating universe will share a similar fate, with distant ones moving apart much faster than c.

A Rapidly Expanding Universe

Something similar is illustrated by the classic example of a universe undergoing extremely rapid, cosmic inflation (though something as dramatic as cosmic inflation isn’t necessary for the general point to hold.) The image of a rubber sheet with ants walking upon it is akin to an inflating universe with photons moving through it. Two photons that pass each other have a constrained relative speed. But nothing prevents two photons far apart from separating from each other faster than c.

In cosmic inflation, the universe doubles in size over a certain time interval T; over the next time interval T, it doubles again; and again, and again, until inflation comes to an end (if it ever does.) Well, that means that the distance L between two typical objects increases faster and faster with every T —and thus the average rate of increase in their distance is itself increasing. After enough steps, the rate at which L increases will inevitably exceed the cosmic speed limit.

This, by the way, is the same reason that the overall distance across the universe can increase faster than the cosmic speed limit, leaving much of the universe beyond our vision. The universe can grow faster than the cosmic speed limit would naively lead one to expect, and so we have no idea how big it might be.

Einstein, however, is not perturbed by this, because this rate of increase of distance is not a relative speed of two objects. Only a relative speed of two objects passing each other, as measured by an observer who is located at one of those objects, need respect the speed limit.

Importantly (as I discussed in this blog post) not even the relative speed of two objects as measured by a third observer need respect the speed limit. Any flash of light that you observe passes you at the cosmic speed limit. So if two flashes of light pass you in opposite directions at the same moment, the relative speed of the two flashes, as you yourself measure them, is just what you would naively guess: twice the cosmic speed limit. Einstein is again unperturbed by this.

In flat space-time, Einstein’s requirement concerning relative speeds extends outward, because in flat space-time, clocks and rulers can be synchronized across space and time. But this cannot be done in sufficiently curved space-time, so Einstein’s requirement remains valid only locally, applying only to objects that are in the same neighborhood.

A final note: in a rapidly contracting universe, the distance to faraway light that is approaching you will decrease faster than c, while the distance to faraway light that is retreating from you may decrease slower or faster than c. It all depends where the light is located, what direction it is heading, and how quickly the universe is contracting. And yet, when the light passes right by your nose, and you can measure its speed with ease and without ambiguity, you will always find that its speed equals c.

28 Responses

  1. I haven’t come across any popular blog, including Sean Caroll’s, that has a main post on space-time foliation; but others have mentioned it in the comments such as here, on your site: https://profmattstrassler.com/2022/05/16/earth-around-the-sun-or-not-the-earth-centered-coordinates-you-should-worry-about/#comment-443905

    I was thinking of a third article on the lines of ‘What [Really] causes gravitational time dilation’ to compliment your other two articles on gravity: ‘The Impossible Commentary: Is Gravity a Force? Is it an Illusion?’ and ‘What [Really] Causes our Twice-Daily Ocean Tides?’. I can barely imagine the time and energy you have to invest in each article, and whether it’s in the interest of you or your publisher, especially since you’ll probably be very busy over the coming months when your book is published.

  2. Do you have plans to write something up on space-time foliation in connection with your series on the speed of light c, space, time etc; do you mention it in your book?

    On the fact of it, it might appear to be too advanced a topic for your high-school/pre-university audience; but it made very crystal clear to me the issues involved with defining a universal space and time here on Earth within its gravitational field. And why stationary clocks ‘apparently’ tick at different rates relative to one another; such as a stationary clock A on the Earth’s surface and another in geostationary orbit above. I don’t see it mentioned in the index of the books from Sean Caroll, Weinberg, Misner/Thorne/Wheeler etc aimed at physics graduates so I doubt it has trickled down into more popular books geared towards us, the interested lay public.

    1. I have no plans to write on such an advanced topic, and indeed I’m not sure I could do a good job on it, as I’m not sufficiently expert. You have to really understand something like that very deeply, backwards and forwards and up and down, to avoid botching it. I’ll leave this to Sean Carroll and his ilk. Are you sure Sean hasn’t done it somewhere? He’s written a lot.

  3. There’s a related tricky “paradox” of measuring superluminal angular speeds for objects traveling with speed close to c but tangentially to the line of sight. This is a real observational fact, for example some relativistic clouds within quasar jets have apparent angular speed many times of c. The effect is maximal when the angle of velocity vector is about 45 deg. For myself, I think of it as follows. If we observe a relativistic cloud which is heading right towards us, the cloud’s “life” itself fast-forwards (blue-shifts) but doesn’t have a chance to influence its angular speed. If the same cloud moves strictly tangentially it’s not boosted in our frame. But if the cloud moves at 45 degs, it’s still boosted and can fast-forward its angular speed as we measure it.

    1. No, not quite. The relativistic model for superluminal jets is given in R. D. Blandford, C. F. McKee & M. J. Rees Nature volume 267, pages 211-216 (1977) and dates back to Rees 1966.

      Suppose that there is a jet pointing towards us with an angle to the line of sight of theta, and it is ejecting blobs at speeds beta = v/c with a Lorentz factor gamma = sqrt(1 / (1 – beta^2)). Then the apparent velocity of the jet, beta_apparent is maximized when sin(theta) = 1/gamma and is beta_max = gamma * beta.

      Since gamma can be > 10 in quasar jets, theta max is typically 5.7 degrees or less. As an example, the quasar 3C273B jet has an beta_apparent of ~10 c, so it is natural to assume that its opening angle is near 6 degrees and its gamma is at least 10.

  4. Possible typo? Without the added text, or something similar, the second half doesn’t follow:

    “Well, that means that the distance L between two typical objects increases [by a factor 2] every T — which in turn means the average rate of increase in their distance is itself increasing.”

    1. Good catch. I had originally said “doubled every T”, and then it would have been obvious. But it doesn’t have to quite double since the objects might have intrinsic motion of their own, which is true in fact in the figure. So I tried to be more careful in my wording, and overdid it.

  5. You mention in flat space-time, clocks and rulers can be synchronized across space and time. But you forgot that in order to synchronize 2 clocks spaced apart , no matter what the distance, you are required to assume that the one-way speed of light is equivalent to the two-way speed. This assumption is only a convention and is called the “Einstein Convention” or the Conventionality of Simultaneity ( as opposed to the relativity of simultaneity ).

    1. Since the laws of nature are invariant under rotations and under shifts of location and time, the speed of light is the same at all times, places and directions; and if the laws of nature were not invariant in this way, then (thanks to Emmy Noether’s famous theorem) experiments would regularly detect violations of energy, momentum and/or angular momentum conservation.

      1. You are correct to assume the speed of light is the same at all times, places and directions; as long as you abide by the “Einstein Convention”.

        1. No, I think you have it backwards.

          When Einstein introduced this, it may have involved a convention. But this is certainly no longer the case; all the invariances of the speed of light are confirmed experimentally. You don’t have to synchronize clocks using Einstein’s approach — you can simply slowly move two synchronized clocks apart. Then you can use them to verify, to the nanosecond, that two photons that are emitted by a stationary positronium atom, going in opposite directions, arrive at the same time.

          By the way the LHC experiments are carefully timed, to the nanosecond, under the assumption of a constant speed of light in all directions.

          1. Again, in your example above with the stationary positronium atom you are working within the context of the Einstein Convention. Because you are assuming to know the value of the one-way speed of light ( = two-way speed ), even though this speed has never been measured and is unmeasurable.

              1. The Einstein Convention, as with any convention, serves as a foundational backdrop for our practice of physics. Indeed, we often treat it as if it were a law rather than a mere convention. This perspective may be reasonable for short distances and brief time periods, but it does not necessarily hold when considering the vast scales of galaxies or beyond.

                1. This paper appears to aim at detecting diurnal fluctuations in the speed of light attributed to Earth’s motion. However, it yields a null outcome, as anticipated. I fail to discern its relevance to the one-way speed of light. Could you explain?

                2. The paper is pretty self-explanatory but there are other ways of measuring the one-way speed of light. For example, an interim target could be placed between the source and the receiver that intercepts a photon as it travels from the source in the direction of the receiver. The frequency of the photon equals its speed divided by its wavelength, f=v/L, (which is true by definition) and the energy and momentum of the photon obey Einstein’s equation E=hf and DE Broglie’s equation L=h/p (which are true by experiment). From these three equations we get v=E/p. And so, measuring the energy and momentum that photons impart to the interim target allows for an easy calculation of v. This is measuring the one-way speed of light since the photon is only traveling one way when it strikes the interim target. This test is easily done in both directions and doesn’t involve using any clocks or rulers. Experimentally, v is measured to be equal to c in countless experiments of these equations and so the one-way speed of light equaling c is settled science.

                3. Experiments like this will work, but they are unnecessary. All sorts of technologies involve sending light in multiple directions and arrangements, so it’s even more than settled; it’s part of the scenery of daily life.

                4. This is an answer to your gedanken experiment involving 2 detectors and the measurements of E and p of the photon. The only way known to measure the momentum of a photon is by Compton scattering. The problem is the photon will as a result be deflected by colliding with an electron.

                5. Farhad, there’s no problem there using Compton scattering but it’s not the only way to measure photon momentum. Photon momentum can be measured using the photo electric effect, pair production, or photon pressure (a good example is light sails). Attach a spring to a light sail and have lots of photons making up the monochromatic light beam strike the light sail. To get the momentum per photon, just divide the total momentum received by the number of photons. Same with energy. Farhad, it seems you have a chip on your shoulder for the constancy postulate. Is that true, and if so, why?

            1. The problem with your assumption however is that it falls short of Occam’s razor in the fashion The Unicorn Behind you. I can assume there’s a unicorn behind me, always behind me, so that I can never see nor measure it. But that requires a universe where there exists a mechanism to ensure this,specifically one centered on me and possibly anyone else who attempts to measure the unicorn.

              The speed of light has been found to the the same in all directions of space by experiment (neglecting things like gravity.) If we want to maintain that light may have a different speed somehow we need a very complex mechanism to hide this from us. A malevolent universe that wishes to defy understanding.

              We all saw the Veritasium video yes,but this is about science not entertainment.Some evidence aside from ‘It could technically be true’ is required.

              1. I think the case is much stronger, and the example of the unicorn is pretty good. There might be a unicorn behind you, but everyone else would see it. Similarly, any conspiracy to hide the behavior of light would be observed in multiple experiments.

                The speed of light is tested million (billions?) of times every day, in all directions and in many locations, by satellites, high-speed traders, particle physics experiments, and GPS receivers. It’s absurd to suggest that there’s still a loophole.

                1. Professor Strassler, you’re right in noting that the speed of light has been measured for a long time, going back to Fizeau or even earlier. However, this doesn’t include the one-way speed of light. I challenge you to present any experiment that has accurately measured the one-way speed. Einstein, ever the conscientious scientist, highlighted the unmeasurable nature of this value, emphasizing the need for a convention to address it.

                2. As though people doing back-and-forth communication don’t do this every day! No more on this topic, you’ve made your incorrect point many times now, and making it more times will convince no one.

              2. Labeling the undefinability of the one-way speed of light as unscientific or merely for entertainment is incorrect. This complex issue was initially introduced by Einstein in 1905, marking its presence from the early days of modern physics. Subsequently, Hans Reichenbach, an eminent philosopher of science, elaborated on this matter, thereby establishing it in its contemporary form.

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